The downpipe is the tube that runs down from the turbocharger. This is where back pressure builds up: the turbine has to push the exhaust gases through a narrowed section and often a catalytic converter. Widening it frees up the turbine, allowing it to build boost earlier and more strongly - it is the exhaust modification you feel most on a turbocharged engine.

What changes behind the wheel: the response. The turbo takes less time to spool up, and the power continues higher in the rev range. On a remapped engine, this is the part that allows the ECU to deliver what the tune promises; without remapping, the ECU compensates and much of the gain remains untapped.